Three positive integers a, b, c with a>c satisfy the following...

Three positive integers

a, b, c

with

a>c

satisfy the following equations:

a c+b+c=b c+a+66, a+b+c=32 .

Find the value of

a

Correct answer

\begin{aligned} & a>c \\ & A, b, c l^{+} \\ & a c+b+c=b c+a+66, \quad a+b+c=32 \\ & \Rightarrow \quad c(a-b)+(b-a)+c=66 \\ & \Rightarrow \quad(a-b)(c-1)+(c-1)=66-1 \\ & \Rightarrow \quad(c-1)(a-b+1)=65=1 \times 65=65 \times 1 \\ & \qquad \qquad \qquad \qquad \qquad \qquad =5 \times 13=13 \times 5\end{aligned}

Case-(i):

c-1=65, a-b+1=1

\Rightarrow \quad c=66

, but

a+b+c=32

\Rightarrow

not possible Case(ii):

c-1=1, a-b+1=65

c=2, \quad a-b=64

a+b+c=32 \Rightarrow a+b=30

\Rightarrow 2 a=64 \Rightarrow a=47, b=-17

not possible Case(iii):

c-1=5, \quad a-b+1=13

\begin{aligned} & c=6, \quad a+\mathrm{i}=26 \\ & a-b=12 \\ & \Rightarrow a=19, \quad b=7 \\ & \text { Case(iv): } \\ & c-1=13, \quad a-b+1=5 \\ & \Rightarrow c-1=13, \quad a-b+1=5 \\ & \Rightarrow c=14, \quad a-b=4 \\ & \quad a=b=18 \\ & \Rightarrow a=11 \text { but } a < c \\ & \Rightarrow \text { only } a=19\end{aligned}